DON'T  WORRY  ABOUT  PEOPLE  STEALING  YOUR  IDEAS.  IF  YOUR  IDEAS  ARE  ANY  GOOD  YOU'LL  HAVE  TO  RAM  THEM  DOWN  PEOPLE'S  THROATS.  —  HOWARD  AIKEN. 

Aiken (1900-1973) was a mathematician and early computer designer (to solve one of his mathematical problems) leading to the Harvard Mark I computer (1944).

Correct answers came from  Delmar Burkitt,  Andrzej Derdzinski,  John Jackson,  Bob Kaplan,  Jeff Lamb,  Alma Litten,  Julia Minturn,  Carla Nuenke,  Carole Perlman,  Amanda Ryal,  Dennis Tomlinson,  Nick Wolf,  Jeff Wolfe,  and  Sally Yocom. 

The second cryptogram was a Haiku structure (3 lines of 5, 7, 5 syllables, respectively) as the first code word indicated.  The answer was

CHAOS  REIGNS  WITHIN.
REFLECT  REPENT  AND  REBOOT.
ORDER  SHALL  RETURN. 

Correct answers came from Andrzej,  Alma,  Julia,  Carla,  Carole,  Nick,  Jeff,  and  Sally. 

In the third puzzle, the order of merit of the six competitors (using initials) was C F A D B E.

Andrzej Derdzinski,  Elizabeth Roote,  Bob Kaplan,  Jeff Lamb,  Dennis Tomlinson,  and  Jeff Wolfe  were correct.

The diagram shown here is an unsuccessful attempt to find a series of four positive whole numbers (circles) with "averageable" squares.  Thus, the average of 161² and 127² is 21025, the square of 145, which is a whole number.  So we write 145 in the rectangle between 161 and 127.  Similarly we can write in 125 and 113.  However, the average of 73² and 97² is 7369, which is not a perfect square.

Your task is to find a ring of numbers in which the four circled numbers, taken by pairs in succession around the ring and squared, will average to a number which is also a perfect square.  There may be multiple possible solutions.  Give me one.